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https://doi.org/10.1038/s41467-020-20148-6 OPEN Floquet spin states in OLEDs

S. Jamali1,6, V. V. Mkhitaryan1,6, H. Malissa 1, A. Nahlawi1, H. Popli1, T. Grünbaum2, S. Bange 2, S. Milster2, ✉ ✉ D. M. Stoltzfus3, A. E. Leung4,5, T. A. Darwish4, P. L. Burn 3, J. M. Lupton 1,2 & C. Boehme1

Electron and hole spins in organic light-emitting diodes constitute prototypical two-level systems for the exploration of the ultrastrong-drive regime of light-matter interactions. Floquet solutions to the time-dependent Hamiltonian of pairs of electron and hole spins reveal that, under non-perturbative resonant drive, when spin-Rabi frequencies become

1234567890():,; comparable to the Larmor frequencies, hybrid light-matter states emerge that enable dipole- forbidden multi-quantum transitions at integer and fractional g-factors. To probe these phenomena experimentally, we develop an electrically detected magnetic-resonance experiment supporting oscillating driving fields comparable in amplitude to the static field defining the Zeeman splitting; and an organic semiconductor characterized by minimal local hyperfine fields allowing the non-perturbative light-matter interactions to be resolved. The experimental confirmation of the predicted Floquet states under strong-drive conditions demonstrates the presence of hybrid light-matter spin excitations at room temperature. These dressed states are insensitive to power broadening, display Bloch-Siegert-like shifts, and are suggestive of long spin coherence times, implying potential applicability for quantum sensing.

1 Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA. 2 Institut für Experimentelle und Angewandte Physik, Universität Regensburg, 93053 Regensburg, Germany. 3 Centre for Organic Photonics & Electronics, School of Chemistry and Molecular Biosciences, The University of Queensland, Brisbane, QLD 4072, Australia. 4 National Deuteration Facility, Australian Nuclear Science and Technology Organization (ANSTO), Lucas Heights, NSW 2234, Australia. 5Present address: Scientific Activities Division, European Spallation Source ERIC, Lund 224 84, Sweden. 6These authors ✉ contributed equally: S. Jamali, V. V. Mkhitaryan. email: [email protected]; [email protected]

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spin in a magnetic field is a perfect discrete-level quan- In principle, magnetic resonances can be resolved down to very tum system, for which resonant electromagnetic radiation small frequencies of a few megahertz, limited only by the overall A fi 5,16 can drive coherent propagation in a well-controlled per- strength of the hyper ne interaction . One advantage of EDMR turbative fashion following Rabi’s theory1. Such propagation is in OLEDs over conventional EPR in radical-pair-based spin-½ used for magnetic resonance applications in spectroscopy and systems17 is that the sample volume can be made almost arbi- imaging, where the thermal-equilibrium population of spin states trarily small. It is therefore possible to achieve high levels of is altered under resonance, inducing an effective magnetization homogeneity both in terms of the static field B0, which defines the change of nuclear or electronic spins2. Experiments are generally Zeeman splitting, and the amplitude of the resonant driving field, performed under the condition that the Zeeman splitting between B1, whereas at the same time penetrating the non-perturbative fi spin states induced by a static magnetic eld is much larger in regime of ultrastrong drive where B1 becomes comparable to B0, energy, or frequency, than the Rabi frequency1. This weak-drive so that the Rabi frequency approaches the Larmor frequency18. limit of resonant pumping is described by perturbation theory. Such an ultrastrong-drive regime is of great interest in con- Magnetic resonance can also be probed by observables secondary temporary condensed-matter physics, although embodiments to magnetization, such as the permutation symmetry of pairs of thereof have proven very challenging to find19,20 and mostly arise spins of electronic excitations. Such spin-dependent transitions in the form of ultrastrong coupling of two-level systems in are reflected in luminescence or conductivity in atomic, mole- resonant optical cavities21–23. cular, or solid-state systems3,4 in optically or electrically detected The breakdown of the perturbative regime of OLED EDMR magnetic resonance (ODMR, EDMR) spectroscopies5. Color has previously been identified under drive conditions of  : centers in crystals, like atomic vacancies in silicon carbide or B1 0 1B0, where conventional power broadening gives way to a diamond, are widely used in ODMR-based quantum metrology variety of strong-drive effects24. Once the driving-field strength and quantum-information processing6, but are limited in one exceeds the inhomogeneous broadening of the individual spins of regard: as dipolar and exchange coupling in the spin pair is the pair induced by the hyperfine fields, the spins become strong, substantial level splitting arises at zero external field, indistinguishable with respect to the radiation and a new set of posing a lower limit on resonance frequency. Such a limitation spin-pair eigenstates is formed25. The resonant field locks the does not exist for weakly coupled spin-½ charge-carrier pairs spin pairs into the triplet configuration, in analogy to the for- which form, for example, by electron transfer in molecular donor- mation of a subradiant state in Dicke’s description of electro- acceptor complexes, where they account for a range of magnetic- magnetic coupling of non-interacting two-level systems26,27. The field effects7. spin-Dicke state manifests itself in EDMR by the appearance of a A particularly versatile way to study weakly coupled spin-½ particular inverted resonance feature25,28. Experimentally, it is pairs is offered by organic light-emitting diodes (OLEDs), which challenging to access this regime of strong and ultrastrong drive generate light from the recombination of electrically injected for the simple reason that very high oscillating magnetic field electrons and holes that bind in pairs of singlet or triplet per- strengths have to be generated in close proximity to the OLED. mutation symmetry8. As the formation rates of intramolecular Using either coils24 or a monolithic microwire integrated in the excitons from intermolecular electron-hole carrier pairs differ for OLED structure18, we were previously able to probe the strong-  : singlet and triplet spin permutations, and singlet pairs tend to drive regime of up to B1 0 3B0. We indeed observed an have shorter lifetimes than triplets, an increase of singlet content inversion of the resonance signal in the device current along with in the electron-hole pair depletes the reservoir of available carriers spectral narrowing occurring owing to the spin-Dicke effect18,24. and leads directly to a change in conductivity9–11. Because carrier Larger oscillating field strengths were not accessible with these spins interact with local hyperfine fields, which originate from earlier experimental setups. In addition, the magnitude of the B0 unresolved hyperfine coupling between charge-carrier spins and field necessary to clearly resolve the resonance was previously the nuclear spins of the ubiquitous protons12, carrier migration limited by the inhomogeneous broadening of the resonance through the active OLED layer gives rise to spin precession and, spectrum due to the hyperfine fields arising from the omnipresent ultimately, mixing of singlet and triplet carrier-pair configura- protons16. Besides limiting spectral resolution, this inhomoge- tions. As a result, OLEDs can exhibit magnetoresistance on neous broadening also constrains the effective coupling strength nanotesla scales at room temperature5. A static magnetic field of the spin states to the driving field25. Qualitatively speaking, the tends to partially suppress this hyperfine spin mixing, an effect disorder increases quantum-mechanical distinguishability of that is reversed under magnetic resonance conditions, which the individual spins with respect to the driving field, lowering the give rise to distinct resonances in the magnetoresistance overall degree of coherence of the incident radiation with the spin functionality13. ensemble that can be achieved under resonant drive25. It is important to stress that such studies are only possible in To date, there has been no formulation of the theoretical materials characterized by very weak spin-orbit coupling14, such expectation of the nature of spin-dependent transitions of two as organic semiconductors, and are not generic to electron-hole weakly coupled spin-½ carriers, electron and hole, under ultra- recombination in LEDs as a whole. In an inorganic LED, for strong resonant drive. Although a Floquet formalism to treat this example, spin mixing occurs by spin-orbit coupling in addition to problem has been put forward previously, this was only pursued the fact that recombination does usually not occur into tightly in the weak-drive limit29. bound, and hence thermally stable, excitonic species. Light gen- In this work, we begin by setting out a strategy for computing eration in inorganic LEDs, in contrast to OLEDs, is therefore such transitions in the ultrastrong-drive regime using the periodic primarily not spin dependent. As such, the subsequent discussion time-dependent spin Hamiltonian in the Floquet formalism. The strictly only applies to OLEDs comprising materials of weak spin- numerically rather detailed calculations can be condensed into a orbit coupling, i.e., of low atomic-order number. The appeal of diagrammatic representation of the resulting hybrid light-matter experimenting with spins in OLEDs is that their coherence time, states, the spin states of the OLED dressed by the driving elec- fi i.e., the transverse spin-relaxation time T2, is only weakly tromagnetic eld. This approach gives us a complete computation dependent on temperature15. This independence is a direct of the EDMR magnetic resonance spectrum of an OLED under consequence of the lack of spin-orbit coupling, which leaves the the condition of ultrastrong drive, with B1 exceeding B0. With carrier dynamics in the local hyperfine fields as the coherence- substantial experimental improvements both in terms of the limiting effect15. material used and with regard to the monolithic OLED-microwire

2 NATURE COMMUNICATIONS | (2021) 12:465 | https://doi.org/10.1038/s41467-020-20148-6 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-20148-6 ARTICLE structure we succeed in experimentally identifying the main referring to the time-independent Fourier component thereof. As predicted Floquet spin states of the OLED under ultrastrong drive described in detail in Supplementary Note 1.2, from this solution, ~ρ conditions. These states are manifested as magnetic-dipole- we can express tr 0 through the following phenomenological forbidden multiple-quantum transitions and resonances at frac- parameters: the rate of spin-pair generation, G; the spin-pair tional g-factors. dissociation rate, rd; the singlet and triplet recombination rates, rS and rT; and the spin-lattice relaxation time, Tsl. For convenience, Results we introduce a characteristic total decay rate of the pair, ¼ þ þ = fi Spin transitions in OLEDs under non-perturbative resonant wd rd rT 1 Tsl, and de ne the difference in singlet and ¼ À drive. Theoretical treatment of the spin dynamics in a strongly triplet recombination rates kr rS rT , which determines the fi jψ i driven electron-hole pair beyond the perturbative regime is overall magnetic- eld response of the conductivity. Using α;0 Π extremely challenging and has not been considered in detail as the eigenvectors of the Floquet Hamiltonian and S as the previously. We approach this problem using quantum- projectionP operator onto the dressed singlet subspace, 30 Π ¼ j ; ih ; j mechanical Floquet theory . The electron-hole spin-pair S n S n S n , we arrive at an expression for the steady- Hamiltonian, H(t), is time dependent owing to the presence of a state spin-density matrix as time-dependent (sinusoidal) driving field. Besides the drive, we X incorporate in H(t) the electron and hole spin coupling to the ~ρ ’ G DE1 : tr 0 ð3Þ fi fi fi 4 α þ ψ jΠ jψ external eld B0 and the effective local internal hyper ne elds, as wd kr α;0 S α;0 well as the isotropic exchange and dipolar interactions between the electron and hole spins. Note that, given the exceedingly weak Equation (3) is derived perturbatively, to leading order in the spin-orbit coupling of the organic semiconductor material used in small difference of singlet and triplet recombination rates kr. Note  jj; jj the experiments discussed in the following, the spin Hamiltonian that kr D J , where D and J are the average dipolar and is perfectly defined without an explicit spin-orbit term, using exchange interactions between electron and hole spins within the effective Landé g-factors instead. We direct the interested reader pairs. Further details of this approximation are discussed in to a recent joint experimental and theoretical examination of the Supplementary Note 1.4. influence of spin-orbit coupling in these materials14. Although all The steady-state device current measured in an experiment is a these interactions are time independent, the local hyperfine fields function of the static and oscillating magnetic fields, which give Δ ðÞ¼; ðÞÀ; ðÞ; and the dipolar interaction are taken to be random and different rise to a change IB0 B1 IB0 B1 IB0 0 of the spin- for different configurations. Further details of the statistical dis- dependent OLED current, i.e., the magnetic resonance. This tribution of spin pairs and the numerical approach to account for steady-state spin-dependent device current probed in experi- ~ρ this are discussed in Supplementary Notes 1.5 and 1.6. ments corresponds to the average of tr 0 over all the random The time-periodic character of the driving field allows us to distributions of local hyperfine fields and dipolar couplings, i.e. write the Fourier decomposition of the Hamiltonian as HαβðÞ¼t P the average over spatial orientations of electronic spin pairs with ðnÞ inωt ω respect to the applied magnetic field, n Hαβ e , where is the angular frequency of the incident radiation. Here, and in the following, the Greek indices α and β I ¼ I tr~ρ ; ð4Þ ; ; 0 0 run over the four spin-pair states, the three triplets Tþ T0 TÀ and the singlet S, which are chosen as the spin Hilbert space basis where I0 is a spin-independent factor that is determined by the for the subsequent discussion. The Floquet dressed states jα; ni conductivity of the OLED. with photon number n ¼ 0; ±1; ±2; ¼ are defined as the We utilize Eqs. (3) and (4) for the numerical calculation of the Δ ðÞ; product of the spin Hilbert space basis state α and the Fourier- EDMR signal IB0 B1 . The Floquet Hamiltonian HF is space basis state |n〉. The dressed states form an orthonormal determined from Eq. (1), for a randomly generated configuration basis in an infinite-dimensional Floquet Hilbert space. A change of local hyperfine fields and dipolar coupling. To approximately fi ψ of n modifies the dressed state while retaining the same spin-pair calculate the in nite-dimensional eigenvectors α;0 of HF entering component. The time-independent, infinite-dimensional Floquet in Eq. (3), we truncate the infinite-dimensional Floquet Hamiltonian HF is then defined in matrix representation as Hamiltonian and the corresponding Hilbert space as described ðÞÀ in Supplementary Note 1.3. The truncation procedure consists of hiα; j jβ; ¼ n m þ ωδ δ : ð1Þ n HF m Hαβ n αβ nm restricting the photon numbers by some integer value, N0,or equivalently restricting the indices m and n in Eq. (1) to run The Floquet approach takes advantage of the fact that HF is between −N and N , where N is determined by the time independent and Hermitian, thus possessing stationary 0 0 0 jψ i ε requirements on the accuracy of the numerical procedure. The eigenvectors, α;n , and real eigenvalues, α;n. The Floquet eigenvectors of the truncated Floquet Hamiltonian are used to Hamiltonian has a periodic structure so that a shift of the indices ~ρ evaluate tr 0 from Eq. (3). The procedure is repeated multiple by the same integer l changes only the diagonal matrix elements times and the field-dependent current is found as the average of hiα; þ j jβ; þ ¼ hiα; j jβ; þ ωδ δ ; the numerical values computed for each random configuration n l HF m l n HF m l αβ nm ð2Þ according to Eq. (4). hα; þ jψ i¼ rendering periodicity of the eigenvectors, n l α;mþl hα; jψ i ε ¼ ε þ ω n α;m , and eigenvalues, α;nþl α;n l . Diagrammatic representation of hybrid light-matter states.In The spin-density matrix of an ensemble of spin pairs, ρ, order to build up an intuitive understanding of the resonant tran- satisfies the stochastic Liouville equation and is used to compute sitions of the electron-hole spin pair emerging under strong and the steady-state singlet content of the spin pair, i.e., the ultrastrong resonance drive and to differentiate the Floquet states observable responsible for the experimentally measured con- responsible for the specific transitions, we develop a diagrammatic ductivity of the OLED. Ultimately, this observable can be representation31 of the dressed states jα; ni.Forsimplicity,we expressed by the trace of the steady-state spin-density matrix, describe the transitions in terms of effective g-factors of resonances. ~ρ tr 0, where the tilde indicates the steady-state solution to the Obviously, this does not relate to the g-factor of the resonant stochastic Liouville equation, which is explicitly time dependent spins—these are all nearly free electrons—but to the ratio between because of the time-periodic Hamiltonian, with the index 0 the driving field oscillation frequency and the frequency

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a | 〉 | 〉 | 〉 | 〉 S, n = n, T–, n = n, T0, n = n, T+, n = n

bcg ≈ 2 g ≈ 1 d g ≈ 4 | 〉 | 〉 | 〉 | 〉 T0, n T+, n – 1 T0, n T+, n – 2 | 〉 | 〉 T–, n T+, n – 1 n n – 1 – 2 n n n – 1

| 〉 | 〉 | 〉 | 〉 n T0, n T–, n + 1 T0, n T–, n + 2 n n + 1 + 2 n n

e

Fig. 1 Multiple-quantum transitions of spin pairs in the singlet-triplet basis. a The Floquet states describing the conductivity of an OLED under magnetic- resonant excitation are defined by the photon number n (illustrated as wavy lines before and after the interaction) and the spin wavefunction (red, green, blue, purple). b The spin-½ resonance of the pair at an effective g-factor of g ≈ 2 corresponds to a raising or lowering of n. c, d Resonances also arise at g ≈ 1 and g ≈ 4 owing to two-photon and half-field transitions, respectively. e Examples of diagrams of the two-photon transitions, where the vortices indicate the creation or annihilation of a photon (•) or spin scattering not involving a photon (×), e.g., owing to hyperfine or dipolar coupling. An infinite number of higher-order loops exists. Magnetoresistance on resonance is calculated by summation over all transitions.

fi corresponding to the Zeeman splitting induced by the static mag- driving eld amplitude B1 for an incident radiation frequency of fi netic eld B0. Figure 1a illustrates the fourfold basis of the Floquet 85 MHz is plotted (see Supplementary Note 1.5 for numerical states. A resonant transition involving the creation or annihilation details). The width of the fundamental resonance g ≈ 2atlow of a photon, shown in Fig. 1b, gives rise to an effective resonance in driving fields is defined by the expectation value of the local ≈ fi fi Δ 25,28 the overall singlet content of the pair at a g-factor g 2. This hyper ne eld experienced by electron or hole spins, Bhyp ,as resonance arises from a superposition of transitions depicted by an described in Supplementary Note 5. The effective g-factors of the infinite number of diagrams of increasing order. Specific examples resonant species are indicated on the left-hand field ordinate. Six of such diagrams describing the g ≈ 1 two-photon resonance owing distinct resonances are seen, with the amplitudes and positions of to simultaneous annihilation or creation of two photons, shown in these depending on B1. The six features are assigned the Floquet- Fig. 1c, are given in Fig. 1e. The vortices of the diagrams mark state transitions marked in the diagram of spin-pair eigenstates in transitions in spin state and occur either by photons, marked as Fig. 2b. Here, states j2i and j3i depend on the particular nature of dots, or due to magnetic scattering interactions such as hyperfine or the intrapair interaction, i.e., dipolar and exchange coupling (see dipolar spin-spin coupling (crosses). The entire singlet content of Supplementary Note 1.1). Three resonances arise from full-field thespinensembleinthesteadystateiscomputedbyapproximating Δm ¼ ±1 transitions, and three from half-field Δm ¼ ±2 tran- the infinite summation over all states (see Supplementary Note 1.4). sitions. Feature (i) is owing to the one-photon spin-½ resonance A further group of transitions (Fig. 1d) involves a change of the and initially undergoes power broadening, before splitting due to magnetic quantum number by Δm ¼ ±2, corresponding to a half- the AC-Zeeman effect and subsequently inverting in amplitude field resonance at g ≈ 4. Since the steady-state singlet content is owing to the spin-Dicke effect24.Theg ≈ 1 feature (ii) results from given by a summation over all spin permutations, resonances at two-photon transitions, and the g ≈ 2/3 feature (iii) from three- fractional g-factors will also arise as combinations of the different photon absorption. Resonances also occur between pure triplet processes. states and result from transitions involving either one (iv) (g ≈ 4), Even though multi-photon transitions in two-level spin three (v) (g ≈ 4/3), or five (vi) (g ≈ 4/5) photons. Features (i–iii) systems have been discussed in the context of electron-spin – clearly move towards lower B0 values with increasing B1, which is resonance spectroscopy32 34, these are not analogous to two- a manifestation of the Bloch-Siegert shift (BSS), discussed in more photon absorption of electric-dipole transitions. As each photon detail below and in Supplementary Note 6. The BSS of the g ≈ 2 carries angular momentum ‘ ¼ 1, a Δ‘ ¼ 2 electric-dipole resonance results in merging into a zero-field resonance at high fi transition requires a nal electronic orbital with different angular B1. The resonances appear self-similar, with the AC-Zeeman and momentum. A spin-½ system can only undergo Δ‘ ¼ 1 spin-Dicke effects apparent in features (i–iii). Crucially, the transitions, however, implying that a magnetic-dipole transition inversion of ΔI at the onset of the spin-Dicke effect coincides with by absorption of two identical photons is impossible. Instead, the formation of a new spin basis of the electromagnetically two-photon magnetic-dipole transitions arise with a combination dressed state24. In this hybrid light-matter state, power broadening of different photons34 whose magnetic-field components have of the resonant transition is absent since the electromagnetic field transversal and longitudinal orientation with regard to the is implicit in the state’s wavefunction25, allowing the BSS to be fi magnetic eld B0. resolved clearly.

Computation of the EDMR spectrum as a function of driving Experimental probing of hybrid light-matter Floquet states in strength. In Fig. 2a the calculated change of spin-dependent OLED EDMR. Identifying these Floquet states experimentally in Δ ð ; Þ current I B0 B1 as a function of Zeeman splitting B0 and the spin-dependent transitions that control the magnetoresistance

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ab10 iii i ii iii 2/3 | 〉 | 〉 | 〉 8 | 〉 | vi 2 3〉 |2〉 | 〉 |3〉 |2〉 1 ii 6 |3〉

(mT) | 〉

v 0 g-factor B vi 4 | 〉 | 〉 2 i v 2 | 〉 | 〉 4 iv 2 |3〉 iv | | 〉 2〉 | 〉 ∞ 0 3 |2〉 | 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 3〉 –1 0 1 | 〉 | 〉 B1 (mT) ΔI (a.u.)

Fig. 2 Floquet spin states in OLED magnetoresistance. a Calculated change of spin-dependent recombination current as a function of static field B0 and oscillating field B1 for a frequency of 85 MHz. b Term diagrams of integer and fractional g-factor multi-photon transitions. (i–iii) Δm ¼ 1 transitions; (iv–vi) Δm ¼ 2 transitions.

of OLEDs poses two challenges. First, the effective hyperfine fields ΔI (nA) 7 must be sufficiently small, such that the different resonances do 0.6 not overlap spectrally. Hyperfine coupling broadens the resonant magnetic-dipole transition inhomogeneously, making individual 6 ii microscopic spins distinguishable in terms of their resonance energy. This disorder determines the threshold field for the 5 onset of spin collectivity24, when each individual spin becomes fi indistinguishable with respect to the driving eld. Previous 4 (mT) studies of the condition of strong magnetic-resonant drive of 0 0 OLEDs employed either conventional hydrogenated organic B i semiconductors18,24, or partially deuterated materials with 3 reduced hyperfine coupling strengths24. To maximize the reso- lution of the experiment, we synthesized a perdeuterated con- 2 iv jugated polymer, poly[2-(2-ethylhexyloxy-d17)-5-methoxy-d3-1,4- phenylenevinylene-d4] (d-MEH-PPV), with 97% of the protons 1 replaced by deuterons35. Second, OLEDs have to be designed with 18 –0.6 integrated microwires which generate the oscillating field . The 0 smaller the OLED pixel relative to the microwire, which narrows 0 50 100 150 200 250 down to a width of 150 μm, the lower the inhomogeneity in both √P (√mW) static and oscillating magnetic fields—at the cost of EDMR signal- to-noise ratio. The alternating current passed through the Fig. 3 Measured change in spin-dependent OLED current ΔI as a function microwire generates heat, requiring careful optimization of elec- of driving power. Thep dominantffiffiffi features of Fig. 2 are indicated by dashed P / B fi P ≤ trical and thermal conductivity of the monolithic OLED- lines. The relationship 1 is con rmed for 100 mW (see microwire device18. Supplementary Note 4), but, due to heating-induced resistivity changes of Figure 3 plots the change ΔI of the steady-state DC forward the microwire generating the resonant field, the uncertainty on the abscissa = scale increases with higher P. current I0 500 nA under 85 MHz RF radiation, as a function of B0 and the square root of the power P of the applied radiation. The dominant Floquet spin state transitions identified in the hyperfine field strength as shown in Supplementary Fig. 6. Note calculation in Fig. 2 are resolved in the experiment and labeled that, because the measurements were actually carried out before correspondingly: the power-broadened g ≈ 2 spin resonance the calculations were completed, unfortunately, we did not probe fi which inverts its sign in the spin-Dicke regime with subsequent the B0 eld region above 7.5 mT, as there was no reason to BSS (i); the two-photon transition and the corresponding BSS of anticipate additional features appearing there. We will leave the g ≈ 1 resonancepffiffiffi (ii); and the half-field resonance at g ≈ 4 (iv). experimental exploration of this interesting field region for future / In principle, P B1, although heating of the microwire at high studies. powers will change the microwire impedance and therefore A simple intuitive rationalization of the intriguing two-photon increase the uncertainty on the abscissa scale in Fig. 3 (see transition can be formulated. Two-photon transitions between Supplementary Note 4). From an analysis of the experimentally the up and down state of a spin-½ species are dipole forbidden. In observed power broadening given in Supplementary Fig. 2, we analogy to light waves, the resonant radiation can be described in estimate that B1 fields of up to 3 mT are reached, consistent with the photon picture. A photon has an angular momentum the universal scaling of EDMR amplitude dependence on B1 with quantum number of 1 and can assume one of the two projection

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ab 15 Experiment B s Theory 8 V)

B0 μ 10 6

 (a.u.) B I 1 4 Δ  5 1 EDMR signal (

≈ B

0

g  2 z y Bhyp Bhyp B1 x 0 0 B hyp -180 -90 0 90 180  (°)

Fig. 4 Intuitive rationalization of the two-photon transition and the angular dependence of the two-photon g ≈ 1 EDMR signal. a Sketch of the magnetic-field configuration acting on a spin within a spin pair. Oscillating magnetic-field components (red arrows) parallel (π) and perpendicular (σ) to the effective local quantizing static magnetic field Bs arise from the decomposition of B1 with respect to the total static magnetic field that results from the superposition of B0 and Bhyp. b Angular dependence of the two-photon EDMR resonance measured at f = 22 MHz (black) for the d-MEH-PPV devices. The experimental results match the sin 2α dependence predicted by theory (red). Lines are a guide to the eye. states of either m ¼þ1orÀ1 relative to the direction of coupling of spins within the spin pairs, which enable the resonances propagation. These so-called σþ and σÀ photons can be under fully parallel and perpendicular excitation. We note that a associated with the right and left-hand circularly polarized fields. slight asymmetry in the angular dependency can be attributed to In a σ-photon state, where the direction of propagation is along additional resonant contributions of the second harmonic of the RF fi fi the quantization axis, the AC magnetic- eld component B1 is signal generated by the RF ampli er. This second harmonic can be = fi perpendicular to B0. The absence of a photon state with m 0 removed by ltering the incident RF radiation using a low-pass can be traced back to a photon having zero mass. However, a filter. Alternatively, we replicate this asymmetry in the calculation in fi k linear AC eld B1 in the direction of quantization, B1 B0, is Fig. 4b by introducing a second harmonic resonant contribution π  : referred to as a -photon propagating perpendicular to the amounting to 0 01 B1. The additional slight asymmetry in the quantization axis. These unconventional π-photons feature in measurement between, e.g., 45° and 135° is also observed for the atomic spectroscopy but are also discussed within a semi-classical fundamental resonance (not shown). This deviation presumably picture of magnetic resonance2,34, and can be thought of as arises from the fact that the RF radiation generated by the stripline photons of total angular momentum zero. Two-photon transi- is not perfectly linearly polarized. tions are only possible with a combination of σ and π-photons. As the excitation scheme employed here would appear to only Discussion σ involve -type photons, it is not immediately obvious how We conclude that the Floquet ansatz presented here to solve the angular momentum is conserved to give rise to the strong two- time-dependent Hamiltonian of a two-level system in the photon resonances observed in theory and experiment. ultrastrong-drive regime provides both an accurate and surpris- Figure 4a provides an intuitive picture for the emergence of the ingly intuitive representation of the complex spin excitations fi two-photon transition. Transversal oscillating magnetic- eld arising under these conditions. These include the spin-Dicke components are orthogonal to B0, but the latter is superimposed state, which manifests a strong BSS, along with dressed light- fi fi with local static isotropic hyper ne elds. This superposition matter states, which support dipole-forbidden multiple-quantum fi leads to the effective static magnetic eld, Bs, tilted from the z axis transitions. Spin-dependent recombination rates between para- fi by a small but nite angle. Thus, the superposition of B0 and the magnetic charge-carrier states in OLEDs offer a remarkably fi fi fi local hyper ne elds results in a total static eld Bs parallel to one versatile testbed to probe this regime of ultrastrong coupling, fi fi component of the oscillating magnetic eld, which is suf cient to visualizing directly the predicted multiple-quantum transitions σ À π enable the two-photon transition conserving angular and fractional g-factor resonances—to our knowledge, better than momentum. To test this picture, in a separate experimental other known quantum systems at room temperature. The iden- setup described in the Methods section, we probed the angular tification of the BSS at room temperature is particularly clear in ≈ dependence of the g 1 resonance. The complete quantum- these solid-state devices compared to any other spectroscopic statistical simulation of the effect using the Floquet ansatz probe of this phenomenon outside of nuclear magnetic reso- α – described above predicts a sin 2 dependence, in agreement with nance36 38. Similarly, direct signatures of hybrid light-matter the geometric arguments forward above. The characteristic Floquet states are observed here to influence spin-dependent features of the prediction from theory are matched by the OLED currents, much like the Floquet states reported in photo- experimental data as shown in Fig. 4b. To allow the angle to electron spectroscopy of topological materials39. be swept, these experiments were performed at a very low B0 of The theoretical results of Fig. 2 provide motivation to extend μ 780 T corresponding to a resonance frequency of 22 MHz. the phase space of the experiment to even higher ratios of Two-photon resonances arise both under parallel and perpendi- = B1 B0. Although experimentally challenging, this is not entirely cular excitation. This observation can be explained by the unfeasible, and may, for example, be conceivable by using illustration in Fig. 4a: photons of both longitudinal (π)and superconducting stripline resonators9. It would be particularly transversal (σ) polarizations exist even for parallel or perpendi- exciting to experimentally identify the spin-Dicke state, which is fi fi cular eld con gurations. The lower B0, the greater the contribu- reflected by the inversion and narrowing of the resonance, in tions of the transversal hyperfine fields and the influence of dipolar the three-photon and two-photon transitions (iii) and (ii)

6 NATURE COMMUNICATIONS | (2021) 12:465 | https://doi.org/10.1038/s41467-020-20148-6 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-20148-6 ARTICLE predicted in Fig. 2. The latter is especially interesting as the BSS the electrical contacts of the thin-film wire RF source and the device electrodes in appears to induce a zero-field resonance, i.e., a resonance order to avoid cross-talk at high RF powers and minimize heating of the OLED feature for B ! 0mTandB  3:7mT. during the room temperature measurements. A photograph of the pixel device 0 1 template is shown in Supplementary Fig. 1a with the pixel located at the image Although we refrain at present from speculating too much on center. An example of the current-voltage characteristics of the device at room possible technological applications of our study, we do note that a temperature is also given in Supplementary Fig. 1b. clear feature in theory and experiment is the demonstration that power broadening is suppressed when the spin-Dicke state is Ultrastrong-drive EDMR spectroscopy. The OLED was subjected to a static fi formed: the one-photon resonance feature (i) power-broadens magnetic eld B0 and irradiated with RF radiation with in-plane amplitude B1 linearly with driving field strength B when B is below the Dicke using an Agilent N5181A frequency generator and an ENI 510 L RF power 1 1 amplifier as illustrated in Supplementary Fig. 1a. At the same time, a steady-state regime, whereas after inversion of the resonance sign (marked in electric current was induced with a V = 2.8 V bias using a Keithley voltage source. red in Figs. 2 and 3) broadening of the inverted feature with a A Stanford Research (SR570) current amplifier was used to detect changes δI of the = further increase in power is minimal. This absence of power steady-state forward current of I0 pffiffiffi500 nA with and without the RF radiation Δ ð ; Þ broadening implies that the hybrid light-matter state, the spin- applied. Measurements of I B0 P as a function of B0 and the square root of Dicke state25, appears to be protected against power broadening the applied microwave power P (which is proportional to B1) were then obtained, as shown in Fig. 3. Further details of the measurement procedure and the deter- precisely because it is a hybrid state. The consequence of this pffiffiffi mination of the P to B1 conversion factor through power broadening are given in protection should be a dramatically enhanced coherence time, the Supplementary Notes 3–7. which in turn may turn out to be useful in quantum magneto- metry applications. Measuring this increased coherence time in Angle-dependent EDMR spectroscopy of the two-photon transition. The 40 fi pulsed experiments at very high B1 could potentially con rm angle-dependence measurements of the two-photon resonance were performed in a this hypothesis. Such experiments would require coherent spin separate low-field setup with different OLED samples, described in detail in refs. 5,42. These OLEDs were much larger, with a pixel area of 3.5 mm2. RF exci- manipulation by RF pulses, which are shorter in duration than tation generated by an Anritsu MG3740A signal generator and amplified by a the RF wave period at the very low Larmor frequencies used here, HUBERT A 1020 RF amplifier was applied to the sample through a coplanar i.e., subcycle magnetic-resonant coherent control, which is tech- stripline designed to match a 50 Ω impedance. In contrast to the measurements on nically feasible with modern pulse synthesizers. the monolithic OLED-microwire structures, these experiments were performed ¼ μ Finally, the work presented here also suggests exciting chal- under constant-current conditions with I 250 A applied by a Keithley 238 source-measure unit, so that the EDMR signal corresponds to a voltage mea- lenges for materials chemists. The linewidth of the resonances in surement. Lock-in detection with a Stanford Research Systems SR830 DSP lock-in the calculated spectra of Fig. 2 is determined primarily from the amplifier and 100% modulation of the RF amplitude at a frequency of 232 Hz was finite inhomogeneous broadening arising from the residual employed. We used a 3D array of Helmholtz coils (Ferronato BH300-3-A), each fi fi supplied by a CAEN ELS Easy-Driver 0520. This allowed us to perform sweeps of hyper ne eld strengths of the deuterated conjugated polymer. By − + fi fi B0 from 2to 2 mT with an arbitrary orientation of the eld vector. Owing to the careful design of materials, it may be possible to reduce hyper ne limited magnetic-field range of the 3D Helmholtz-coil set, we resorted to a lower π  μ coupling even further, for example, by extending the -electron excitation frequency of 22 MHz with B1 200 T. system in polycyclic aromatic hydrocarbons to lower the inter- action between electronic and nuclear magnetic moments. With Data availability such advances, it will be possible to resolve resonances at even The raw data that support the plots within this paper and the other findings of this study fi lower B0 elds and therefore reach even higher effective ratios of are available from the corresponding authors upon reasonable request. B1 to B0. Code availability The Fortran code used for the numerical simulations discussed in this paper is available Methods from the corresponding authors upon reasonable request. Details of the computational procedure, including the modifications made to the conventional spin-pair model of spin-dependent recombination in an OLED28,41, are given in Supplementary Note 1.5 along with a list of the model Received: 18 June 2020; Accepted: 28 October 2020; parameters used.

Device fabrication. In order to establish and detect ultrastrong electron-spin magnetic resonance drive conditions, we developed a new monolithic OLED device References structure with an integrated RF microwire. These small circular (57 µm diameter) 1. Rabi, I. I. Space quantization in a gyrating magnetic field. Phys. Rev. 51, single-pixel OLEDs allow for bipolar charge-carrier injection using electron and 652–654 (1937). fi hole injector layers on top of an electrically and thermally separated thin- lm wire 2. Schweiger, A. & Jeschke, G. Principles of Pulse Electron Paramagnetic fi capable of producing RF magnetic elds B1 when connected to an RF source. The Resonance. (Oxford University Press, 2001). μ wire itself runs across the substrate and is shrunk down to 150 m width beneath 3. Geschwind, S., Collins, R. J. & Schawlow, A. L. Optical detection of the OLED pixel. Details of the layer sequence necessary to achieve electrical and 3+ 18 paramagnetic resonance in an excited state of Cr in Al2O3. Phys. Rev. Lett. thermal decoupling are given in ref. . Further details can be found in Supple- 3, 545–548 (1959). mentary Note 2. The active polymer, d-MEH-PPV, was dissolved in toluene at a – 4. Wrachtrup, J., von Borczyskowski, C., Bernard, J., Orrit, M. & Brown, R. concentration of 4.5 g/L at a temperature of 50 70 °C and deposited by spin-casting Optical detection of magnetic resonance in a single molecule. Nature 363, at 800 rpm, as described in ref. 35. The layer was sandwiched as a 100 nm thin film 244–245 (1993). between a Ca/Al stack for electron injection and a TiO (3 nm)/Au (80 nm)/Ti 2 5. Grünbaum, T. et al. OLEDs as models for bird magnetoception: detecting (10 nm) layer covered with PEDOT:PSS (Ossila Al 4083) for hole injection. The electron spin resonance in geomagnetic fields. Faraday Discuss. 221, 92 (2020). structure is comparable to that used in previous studies of the strong-drive regime, π 6. Doherty, M. W. et al. The nitrogen-vacancy colour centre in diamond. Phys. where a different -conjugated polymer with much larger resonance linewidths was – used24. For the study presented here, the active-layer material, the layer stack, and Rep. 528,1 45 (2013). the pixel device geometry were redesigned and optimized in order to allow for 7. van Schooten, K. J., Baird, D. L., Limes, M. E., Lupton, J. M. & Boehme, C. Probing long-range carrier-pair spin-spin interactions in a conjugated larger B1/B0 ratios to be reached. The monolithic single-pixel OLED device was made following a preparation and deposition protocol as illustrated in Jamali polymer by detuning of electrically detected spin beating. Nat. Commun. 6, et al.18 with the following changes: (i) the active polymer layer where the electron- 6688 (2015). hole pair recombination takes place was nominally 100 nm thick, consisting of 8. Pope, M. & Swenberg, C. E. Electronic Processes in Organic Crystals and spin-coated perdeuterated d-MEH-PPV; (ii) the sample template preparation Polymers. (Oxford University Press, New York, 1999). included placing the silicon wafer in a dry thermal oxidation furnace at 1000 °C for 9. Wang, J., Chepelianskii, A., Gao, F. & Greenham, N. C. Control of exciton 78 min, producing a 50 nm SiO2 layer on top of the Si wafer for insulation and spin statistics through spin polarization in organic optoelectronic devices. Nat. better adhesion of the subsequent layer (amorphous SiN); and (iii) an entirely Commun. 3, 1191 (2012). different lateral layout of the templates (cf. Supplementary Fig. 1a) was used 10. Kraffert, F. et al. Charge separation in PCPDTBT:PCBM blends from an EPR compared to that described in Jamali et al.18, providing larger separation between perspective. J. Phys. Chem. C 118, 28482–28493 (2014).

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